1899-1900.] Dr Knott on Kleins View of Quaternions. 
27 
stage ; for, as they admit on p. 66, they “ have no occasion in 
succeeding chapters to return to quaternion calculation.’ 5 
Meanwhile, having asserted the identity of quaternion and 
rotation, the authors proceed to adopt Hamilton’s nomenclature, 
calling D the scalar part and ( iA +jB + TcC) the vector part of the 
quaternion (Drehstreckung 1 ?). 
They then consider a quaternion which is reduced to its vector 
part, and which is identified with a Drehstreckung whose angle of 
rotation is to = 7 r, that is, two right angles. This special kind of 
Drehstreckung, this semi-revolution about an axis, combined with 
isotropic expansion, is called a Wendestreckung . Regarded as a 
Wendestreckung the vector is assumed to take the analytical form 
V=iX+jY+kZ 
But if this be a Wendestreckung, so also is the quantity 
iA +jB + kC, which, on their assumptions, is an important part 
of the Drehstreckung Q. This no doubt is the Wendestreckung 
to which the Drehstreckung Q is reduced when o> = 7r. But, 
when associated with the so-called scalar in the complete expression 
for the Drehstreckung , the so-called vector cannot he interpreted 
in any sense as a Wendestreckung. The most elementary con- 
siderations in the geometry of rotations show that, in its effect 
upon a body, the assumed analytical expression for the Dreh- 
streckung must be treated as a whole. The expression, in fact, 
is non-distributive. Thus v(iA +jB + kC + D), where v is a vector 
line and the part in brackets a Drehstreckung , cannot be expanded 
in the form viA + vjB -1- vkC + vD. Nevertheless the authors assert 
(p. 59) that two quantities of the form Q may be added together 
as Hamiltonian quaternions are added — i.e., the distributive law, 
which holds for true quaternions, is assumed to hold also for 
Drehstreckungen. But this assumption is inadmissible ; for, as a 
matter of fact (see Professor Tait’s foregoing paper, p. 23), two 
Drehstreckungen when added together cannot in general be 
represented as a single Drehstreckung. 
Throughout pp. 59-62 the quantities of the form Q and V are 
treated analytically exactly as Hamilton’s quaternions and vectors 
are treated. Thus, in order that the magnitudes A" B" G" D" 
which constitute Q" ( = QQ') may be properly related to the 
